MATHEMATICAL METHODS FOR PHYSICS
Learning outcomes of the course unit
The course is intended to complete a basic training in mathematical analysis, giving the necessary background for the courses in physics of the first three years
Introductory courses in analysis, algebra and geometry.
Course contents summary
1. Differential Equations in the Complex Field
1.1 - Regular and singular points of equations. Uniformity.
1.2 - Fuchs’ Theorem
1.3 - Equations with three singular points. Riemann’s symbol.
1.3 - Gauss Equation. Hypergeometric function and series.
1.5 - Confluent hypergeometric equantion.
1.6 - Legendre Equation and polynomials. Spherical harmonics.
1.7 - Hermite and La guerre Equations and polynomials.
1.8 - Bessel Equations and functions.
1.9 - Applications to the Schroedinger’s equations.
2. Approximation by functions
2.1 - Approximation criteria.
2.2 - Orthogonalization.
2.3 - Classical polynomials.
2.4 - Trigonometric and exponential series.
2.5 - L1 and L2 spaces, general properties.
3. Integral Representations.
3.1 - The Fourier Integral and its L1 properties.
3.2 - Plancherel Thgeorem, L2 properties.
3.3 - The Laplace Integral and its properties.
3.4 - Inversion of the Laplace Transform.
3.5 - Applications to partila derivatives equations.
4. Outlines in Measure Theory and Integration.
4.1 – Families of sets, rings, algebras and sigma algebras.
4.2 – Lebesgue Measure
4.3 – Lebesgue Integral.
4.4 – Absolutely continuous and singular functions.
V. Smirnov: Corso di Matematica Superiore, vol. III,2 Editori Riuniti 1982
M. R. Spiegel: Trasformata di Laplace Etas, collana Schaum
A. Kolmogorov e S Fomin : Analisi Funzionale Mir
C. Bernardini, O. Ragnisco, P.M. Santini: Metodi Matematici della Fisica, NIS
Frontal lectures and exercises
Final written and oral examination